The electrostatic interaction between two parallel charged cylinders of arbitrary diameters alpha(1) and alpha(2) is studied by solving numerically the nonlinear Poisson-Boltzmann equation. A limit of particular interest is that of a charged cylinder interacting with a charged plane (alpha(1) --> infinity). The interaction free energy is calculated, as a function of the separation distance between the cylinders, for either fixed charge density or mixed (fixed charge, fixed potential) boundary conditions.